Perturbations of roots under linear transformations of polynomials

Abstract

Let \cP_n be the complex vector space of all polynomials of degree at most $n$. We give several characterizations of the linear operators T\in\cL(\cP_n) for which there exists a constant $C > 0$ such that for all nonconstant p\in\cP_n there exist a root $u$ of $p$ and a root $v$ of $Tp$ with $|u-v|\leq C$. We prove that such perturbations leave the degree unchanged and, for a suitable pairing of the roots of $p$ and $Tp$, the roots are never displaced by more than a uniform constant independent on $p$. We show that such ``good'' operators $T$ are exactly the invertible elements of the commutative algebra generated by the differentiation operator. We provide upper bounds in terms of $T$ for the relevant constants.Comment: 23 page